Reference for: Every element of a finite field is the sum of two squares.

Question

Note: This is a reference-request question. It doesn't require the usual level of context. I am not asking for a proof.

It is well known that

Theorem: Let $x\in F$ for a finite field $F$. Then there exist $a,b\in F$ such that $x=a^2+b^2$.

I need the theorem for my dissertation. Therefore, I need a reference, please.

For context: I am writing up this (with a citation and the appropriate credit, of course), which a theorem of mine in a draft of my dissertation now relies upon.

Answer 1

Proposition 1 in Vitaly Bergelson, Andrew Best, and Alex Iosevich, Sums of Powers in Large Finite Fields: A Mix of Methods, The American Mathematical Monthly, 128:8, pp. 701-718. This has the advantage of being open access.

Answer 2

L'Arithmétique des corps, by Paulo Ribenboim.

Page 182, chapitre X:"Sommes des carrés"

Hermann París. Collection Méthodes. 1972.

There is an English translation.